Property $T$ of reduced $C^{\ast }$-crossed products by discrete groups

Abstract

We generalize the main result of Kamalov and show that if $G$ is an amenable discrete group with an action $\alpha$ on a finite nuclear unital $C^{\ast }$-algebra $A$ such that the reduced crossed product $A{⋊}_{\alpha ,r}G$ has property $T$, then $G$ is finite and $A$ is finite dimensional. As an application, an infinite discrete group $H$ is non-amenable if and only if the uniform Roe algebra $C_u^{\ast }(H)$ has property $T$.

References in corpus (2)

• Kazhdan's Property T and C*-algebras
• Property $(T)$ and strong Property $(T)$ for unital $C^{\ast }$-algebras